HSC Extension 1 and 2 Mathematics/3-Unit/Preliminary/Harder 2-Unit

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Implicit differentiation[edit]

Implicit differentiation is a method of differentiating an expression in x and y, where x and y are related in some manner and neither are constant.

For example, one could differentiate f(x) = y^2 with respect to x as follows:

Using the chain rule:
\tfrac{df}{dx} & = \tfrac{df}{dy} \times \tfrac{dy}{dx} \\
               & = 2y \times \tfrac{dy}{dx}

It is useful to think of implicit differentiation as normal differentiation with respect to x, only whenever you come across a term with y, you multiply the differentiated term by dy/dx.

Another example: find the derivative dy/dx of x^2+y^2=r^2


2x+2y.\frac{dy}{dx} & = 0 \\
2x                   & = -2y.\frac{dy}{dx} \\
\frac{dy}{dx}       & = -\frac{x}{y}